Statistical Modelling of COVID-19 Outbreak in Italy

16 Apr 2020



Nonlinear growth models

Nonlinear growth models represent an instance of nonlinear regression models, a class of models taking the general form \[ y = \mu(x, \theta) + \epsilon, \] where \(\mu(x, \theta)\) is the mean function which depends on a possibly vector-valued parameter \(\theta\), and a possibly vector-valued predictor \(x\). The stochastic component \(\epsilon\) represents the error with mean zero and constant variance. Usually, a Gaussian distribution is also assumed for the error term.

By defining the mean function \(\mu(x, \theta)\) we may obtain several different models, all characterized by the fact that parameters \(\theta\) enter in a nonlinear way into the equation. Parameters are usually estimated by nonlinear least squares which aims at minimizing the residual sum of squares.

Exponential

\[ \mu(x) = \theta_1 \exp\{\theta_2 x\} \] where \(\theta_1\) is the value at the origin (i.e. \(\mu(x=0)\)), and \(\theta_2\) represents the (constant) relative ratio of change (i.e. \(\frac{d\mu(x)}{dx }\frac{1}{\mu(x)} = \theta_2\)). Thus, the model describes an increasing (exponential growth if \(\theta_2 > 0\)) or decreasing (exponential decay if \(\theta_2 < 0\)) trend with constant relative rate.

Logistic

\[ \mu(x) = \frac{\theta_1}{1+\exp\{(\theta_2 - x)/\theta_3\}} \] where \(\theta_1\) is the upper horizontal asymptote, \(\theta_2\) represents the x-value at the inflection point of the symmetric growth curve, and \(\theta_3\) represents a scale parameter (and \(1/\theta_3\) is the growth-rate parameter that controls how quickly the curve approaches the upper asymptote).

Gompertz

\[ \mu(x) = \theta_1 \exp\{-\theta_2 \theta_3^x\} \] where \(\theta_1\) is the horizontal asymptote, \(\theta_2\) represents the value of the function at \(x = 0\) (displacement along the x-axis), and \(\theta_3\) represents a scale parameter.

The difference between the logistic and Gompertz functions is that the latter is not symmetric around the inflection point.

Richards

\[ \mu(x) = \theta_1 (1 - \exp\{-\theta_2 x\})^{\theta_3} \] where \(\theta_1\) is the horizontal asymptote, \(\theta_2\) represents the rate of growth, and \(\theta_3\) in part determines the point of inflection on the y-axis.

Data

Dipartimento della Protezione Civile: COVID-19 Italia - Monitoraggio della situazione http://arcg.is/C1unv

Source: https://github.com/pcm-dpc/COVID-19

url = "https://raw.githubusercontent.com/pcm-dpc/COVID-19/master/dati-andamento-nazionale/dpc-covid19-ita-andamento-nazionale.csv"
COVID19 <- read.csv(file = url, stringsAsFactors = FALSE)
COVID19$data <- as.Date(COVID19$data)
# DT::datatable(COVID19)

Warnings

- 29/03/2020: dati Regione Emilia Romagna parziali (dato tampone non aggiornato).
- 26/03/2020: dati Regione Piemonte parziali (-50 deceduti - comunicazione tardiva)
- 18/03/2020: dati Regione Campania non pervenuti.
- 18/03/2020: dati Provincia di Parma non pervenuti.
- 17/03/2020: dati Provincia di Rimini non aggiornati
- 16/03/2020: dati P.A. Trento e Puglia non pervenuti.
- 11/03/2020: dati Regione Abruzzo non pervenuti.
- 10/03/2020: dati Regione Lombardia parziali.
- 07/03/2020: dati Brescia +300 esiti positivi


Modelling total infected

# create data for analysis
data = data.frame(date = COVID19$data,
                  y = COVID19$totale_casi,
                                    dy = reldiff(COVID19$totale_casi))
data$x = as.numeric(data$date) - min(as.numeric(data$date)) + 1
DT::datatable(data, options = list("pageLength" = 5))

Estimation

Exponential

mod1_start = lm(log(y) ~ x, data = data)
b = unname(coef(mod1_start))
start = list(th1 = exp(b[1]), th2 = b[2])
exponential <- function(x, th1, th2) th1 * exp(th2 * x)
mod1 = nls(y ~ exponential(x, th1, th2), data = data, start = start)
summary(mod1)
## 
## Formula: y ~ exponential(x, th1, th2)
## 
## Parameters:
##        Estimate  Std. Error t value       Pr(>|t|)    
## th1 12262.24093  1536.51799   7.981 0.000000000156 ***
## th2     0.05261     0.00278  18.926        < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 15600 on 51 degrees of freedom
## 
## Number of iterations to convergence: 12 
## Achieved convergence tolerance: 0.000004541

Logistic

mod2 = nls(y ~ SSlogis(x, Asym, xmid, scal), data = data)
summary(mod2)
## 
## Formula: y ~ SSlogis(x, Asym, xmid, scal)
## 
## Parameters:
##         Estimate  Std. Error t value Pr(>|t|)    
## Asym 172301.4694   2238.5230   76.97   <2e-16 ***
## xmid     33.4506      0.2795  119.69   <2e-16 ***
## scal      7.1184      0.1785   39.87   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3083 on 50 degrees of freedom
## 
## Number of iterations to convergence: 0 
## Achieved convergence tolerance: 0.000002445

Gompertz

mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data)
# start = list(Asym = coef(mod2)[1])
# tmp = list(y = log(log(start$Asym) - log(data$y)), x = data$x)
# b = unname(coef(lm(y ~ x, data = tmp)))
# start = c(start, c(b2 = exp(b[1]), b3 = exp(b[2])))
# mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data, start = start,
#            control = nls.control(maxiter = 1000))
summary(mod3)
## 
## Formula: y ~ SSgompertz(x, Asym, b2, b3)
## 
## Parameters:
##            Estimate     Std. Error t value Pr(>|t|)    
## Asym 206759.9968046   1924.3186126   107.4   <2e-16 ***
## b2        9.4041263      0.2000748    47.0   <2e-16 ***
## b3        0.9310767      0.0009183  1013.9   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1154 on 50 degrees of freedom
## 
## Number of iterations to convergence: 0 
## Achieved convergence tolerance: 0.000006813

Richards

richards <- function(x, th1, th2, th3) th1*(1 - exp(-th2*x))^th3
Loss  <- function(th, y, x) sum((y - richards(x, th[1], th[2], th[3]))^2) 
start <- optim(par = c(coef(mod2)[1], 0.001, 1), fn = Loss, 
               y = data$y, x = data$x)$par
names(start) <- c("th1", "th2", "th3")
mod4 = nls(y ~ richards(x, th1, th2, th3), data = data, start = start,
           # trace = TRUE, algorithm = "plinear", 
           control = nls.control(maxiter = 1000, tol = 0.1))
# algorithm is not converging... 
summary(mod4)
## 
## Formula: y ~ richards(x, th1, th2, th3)
## 
## Parameters:
##          Estimate    Std. Error t value Pr(>|t|)    
## th1 218093.601139   3026.646050   72.06   <2e-16 ***
## th2      0.061489      0.001382   44.49   <2e-16 ***
## th3      6.690620      0.222148   30.12   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1285 on 50 degrees of freedom
## 
## Number of iterations to convergence: 3 
## Achieved convergence tolerance: 0.01543
# library(nlmrt)
# mod4 = nlxb(y ~ th1*(1 - exp(-th2*x))^th3, 
#             data = data, start = start, trace = TRUE)

Models comparison

models = list("Exponential model" = mod1, 
              "Logistic model" = mod2, 
              "Gompertz model" = mod3,
              "Richards model" = mod4)
tab = data.frame(loglik = sapply(models, logLik),
                 df = sapply(models, function(m) attr(logLik(m), "df")),
                 Rsquare = sapply(models, function(m) 
                                  cor(data$y, fitted(m))^2),
                 AIC = sapply(models, AIC),
                 AICc = sapply(models, AICc),
                 BIC = sapply(models, BIC))
sel <- apply(tab[,4:6], 2, which.min)
tab$"" <- sapply(tabulate(sel, nbins = length(models))+1, symnum,
                 cutpoints = 0:4, symbols = c("", "*", "**", "***"))
knitr::kable(tab)
loglik df Rsquare AIC AICc BIC
Exponential model -585.9084 3 0.9398151 1177.8168 1178.3066 1183.7277
Logistic model -499.4427 4 0.9977156 1006.8855 1007.7188 1014.7666
Gompertz model -447.3792 4 0.9996429 902.7583 903.5917 910.6395 ***
Richards model -453.0506 4 0.9996001 914.1013 914.9346 921.9824 
ggplot(data, aes(x = date, y = y)) + 
  geom_point() +
  geom_line(aes(y = fitted(mod1), color = "Exponential")) +
  geom_line(aes(y = fitted(mod2), color = "Logistic")) +
  geom_line(aes(y = fitted(mod3), color = "Gompertz")) +
  geom_line(aes(y = fitted(mod4), color = "Richards")) +
  labs(x = "", y = "Infected", color = "Model") +
  scale_color_manual(values = cols) +
  scale_y_continuous(breaks = seq(0, coef(mod2)[1], by = 10000),
                     minor_breaks = seq(0, coef(mod2)[1], by = 5000)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

last_plot() +
  scale_y_continuous(trans = "log10", limits = c(100,NA)) +
  labs(y = "Infected (log10 scale)")

Predictions

Point estimates

df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1),
               fit1 = predict(mod1, newdata = df),
               fit2 = predict(mod2, newdata = df),
               fit3 = predict(mod3, newdata = df),
               fit4 = predict(mod4, newdata = df))
ylim = c(0, max(df[,c("fit2", "fit3")]))
ggplot(data, aes(x = date, y = y)) + 
  geom_point() +
  geom_line(data = df, aes(x = date, y = fit1, color = "Exponential")) +
  geom_line(data = df, aes(x = date, y = fit2, color = "Logistic")) +
  geom_line(data = df, aes(x = date, y = fit3, color = "Gompertz")) +
  geom_line(data = df, aes(x = date, y = fit4, color = "Richards")) +
  coord_cartesian(ylim = ylim) +
  labs(x = "", y = "Infected", color = "Model") +
  scale_y_continuous(breaks = seq(0, max(ylim), by = 10000),
                     minor_breaks = seq(0, max(ylim), by = 5000)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  scale_color_manual(values = cols) +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

Prediction intervals

# compute prediction using Moving Block Bootstrap (MBB) for nls
df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1))

pred1 = cbind(df, "fit" = predict(mod1, newdata = df))
pred1[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod1, df[df$x > max(data$x),])[,2:3]

pred2 = cbind(df, "fit" = predict(mod2, newdata = df))
pred2[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod2, df[df$x > max(data$x),])[,2:3]

pred3 = cbind(df, "fit" = predict(mod3, newdata = df))
pred3[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod3, df[df$x > max(data$x),])[,2:3]

pred4 = cbind(df, "fit" = predict(mod4, newdata = df))
pred4[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod4, df[df$x > max(data$x),])[,2:3]

# predictions for next day
pred = rbind(subset(pred1, x == max(data$x)+1, select = 2:5),
             subset(pred2, x == max(data$x)+1, select = 2:5),
             subset(pred3, x == max(data$x)+1, select = 2:5),
             subset(pred4, x == max(data$x)+1, select = 2:5))
print(pred, digits = 3)
##           date    fit    lwr    upr
## 54  2020-04-17 210040 165753 252336
## 541 2020-04-17 163202 155454 169806
## 542 2020-04-17 169475 166646 172301
## 543 2020-04-17 170486 167470 174026

ylim = c(0, max(pred2$upr, pred3$upr, na.rm=TRUE))
ggplot(data, aes(x = date, y = y)) + 
  geom_point() +
  geom_line(data = pred1, aes(x = date, y = fit, color = "Exponential")) +
  geom_line(data = pred2, aes(x = date, y = fit, color = "Logistic")) +
  geom_line(data = pred3, aes(x = date, y = fit, color = "Gompertz")) +
  geom_line(data = pred4, aes(x = date, y = fit, color = "Richards")) +
  geom_ribbon(data = pred1, aes(x = date, ymin = lwr, ymax = upr), 
              inherit.aes = FALSE, fill = cols[1], alpha=0.3) +
  geom_ribbon(data = pred2, aes(x = date, ymin = lwr, ymax = upr), 
              inherit.aes = FALSE, fill = cols[2], alpha=0.3) +
  geom_ribbon(data = pred3, aes(x = date, ymin = lwr, ymax = upr),
              inherit.aes = FALSE, fill = cols[3], alpha=0.3) +
  geom_ribbon(data = pred4, aes(x = date, ymin = lwr, ymax = upr),
              inherit.aes = FALSE, fill = cols[4], alpha=0.3) +
  coord_cartesian(ylim = c(0, max(ylim))) +
  labs(x = "", y = "Infected", color = "Model") +
  scale_y_continuous(minor_breaks = seq(0, max(ylim), by = 10000)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  scale_color_manual(values = cols) +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

Modelling total deceased

# create data for analysis
data = data.frame(date = COVID19$data,
                  y = COVID19$deceduti,
                                    dy = reldiff(COVID19$deceduti))
data$x = as.numeric(data$date) - min(as.numeric(data$date)) + 1
DT::datatable(data, options = list("pageLength" = 5))

Estimation

Exponential

mod1_start = lm(log(y) ~ x, data = data)
b = unname(coef(mod1_start))
start = list(th1 = exp(b[1]), th2 = b[2])
exponential <- function(x, th1, th2) th1 * exp(th2 * x)
mod1 = nls(y ~ exponential(x, th1, th2), data = data, start = start)
summary(mod1)
## 
## Formula: y ~ exponential(x, th1, th2)
## 
## Parameters:
##        Estimate  Std. Error t value      Pr(>|t|)    
## th1 1069.698020  150.473233   7.109 0.00000000366 ***
## th2    0.060222    0.003059  19.684       < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1891 on 51 degrees of freedom
## 
## Number of iterations to convergence: 12 
## Achieved convergence tolerance: 0.000009064

Logistic

mod2 = nls(y ~ SSlogis(x, Asym, xmid, scal), data = data)
summary(mod2)
## 
## Formula: y ~ SSlogis(x, Asym, xmid, scal)
## 
## Parameters:
##        Estimate Std. Error t value Pr(>|t|)    
## Asym 22839.6491   278.0279   82.15   <2e-16 ***
## xmid    36.0636     0.2348  153.60   <2e-16 ***
## scal     6.5345     0.1446   45.19   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 335.4 on 50 degrees of freedom
## 
## Number of iterations to convergence: 0 
## Achieved convergence tolerance: 0.000002959

Gompertz

mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data)
# manually set starting values
# start = list(Asym = coef(mod2)[1])
# tmp = list(y = log(log(start$Asym) - log(data$y)), x = data$x)
# b = unname(coef(lm(y ~ x, data = tmp)))
# start = c(start, c(b2 = exp(b[1]), b3 = exp(b[2])))
# mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data, start = start, 
#            control = nls.control(maxiter = 10000))
summary(mod3)
## 
## Formula: y ~ SSgompertz(x, Asym, b2, b3)
## 
## Parameters:
##           Estimate    Std. Error t value Pr(>|t|)    
## Asym 28125.3224339   241.1227794  116.64   <2e-16 ***
## b2      13.4738592     0.2907493   46.34   <2e-16 ***
## b3       0.9274850     0.0008245 1124.95   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 117.9 on 50 degrees of freedom
## 
## Number of iterations to convergence: 0 
## Achieved convergence tolerance: 0.000001661

Richards

richards <- function(x, th1, th2, th3) th1*(1 - exp(-th2*x))^th3
Loss  <- function(th, y, x) sum((y - richards(x, th[1], th[2], th[3]))^2) 
start <- optim(par = c(coef(mod2)[1], 0.001, 1), fn = Loss, 
               y = data$y, x = data$x)$par
names(start) <- c("th1", "th2", "th3")
mod4 = nls(y ~ richards(x, th1, th2, th3), data = data, start = start,
           # trace = TRUE, algorithm = "port", 
           control = nls.control(maxiter = 1000))
summary(mod4)
## 
## Formula: y ~ richards(x, th1, th2, th3)
## 
## Parameters:
##         Estimate   Std. Error t value Pr(>|t|)    
## th1 29214.826661   354.504718   82.41   <2e-16 ***
## th2     0.068143     0.001226   55.58   <2e-16 ***
## th3    10.383636     0.333519   31.13   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 135.5 on 50 degrees of freedom
## 
## Number of iterations to convergence: 6 
## Achieved convergence tolerance: 0.00000413

Models comparison

models = list("Exponential model" = mod1, 
              "Logistic model" = mod2, 
              "Gompertz model" = mod3,
              "Richards model" = mod4)
tab = data.frame(loglik = sapply(models, logLik),
                 df = sapply(models, function(m) attr(logLik(m), "df")),
                 Rsquare = sapply(models, function(m) 
                                  cor(data$y, fitted(m))^2),
                 AIC = sapply(models, AIC),
                 AICc = sapply(models, AICc),
                 BIC = sapply(models, BIC))
sel <- apply(tab[,4:6], 2, which.min)
tab$"" <- sapply(tabulate(sel, nbins = length(models))+1, symnum,
                 cutpoints = 0:4, symbols = c("", "*", "**", "***"))
knitr::kable(tab)
loglik df Rsquare AIC AICc BIC
Exponential model -474.0555 3 0.9492970 954.1110 954.6008 960.0219
Logistic model -381.8797 4 0.9984589 771.7593 772.5926 779.6405
Gompertz model -326.4521 4 0.9997838 660.9043 661.7376 668.7854 ***
Richards model -333.8191 4 0.9997303 675.6383 676.4716 683.5194 
ggplot(data, aes(x = date, y = y)) + 
  geom_point() +
  geom_line(aes(y = fitted(mod1), color = "Exponential")) +
  geom_line(aes(y = fitted(mod2), color = "Logistic")) +
  geom_line(aes(y = fitted(mod3), color = "Gompertz")) +
  geom_line(aes(y = fitted(mod4), color = "Richards")) +
  labs(x = "", y = "Deceased", color = "Model") +
  scale_color_manual(values = cols) +
  scale_y_continuous(breaks = seq(0, coef(mod2)[1], by = 1000),
                     minor_breaks = seq(0, coef(mod2)[1], by = 500)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

last_plot() +
  scale_y_continuous(trans = "log10", limits = c(10,NA)) +
  labs(y = "Deceased (log10 scale)")

Predictions

Point estimates

df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1),
               fit1 = predict(mod1, newdata = df),
               fit2 = predict(mod2, newdata = df),
               fit3 = predict(mod3, newdata = df),
               fit4 = predict(mod4, newdata = df))
ylim = c(0, max(df[,-(1:3)]))
ggplot(data, aes(x = date, y = y)) + 
  geom_point() +
  geom_line(data = df, aes(x = date, y = fit1, color = "Exponential")) +
  geom_line(data = df, aes(x = date, y = fit2, color = "Logistic")) +
  geom_line(data = df, aes(x = date, y = fit3, color = "Gompertz")) +
  geom_line(data = df, aes(x = date, y = fit4, color = "Richards")) +
  coord_cartesian(ylim = ylim) +
  labs(x = "", y = "Deceased", color = "Model") +
  scale_y_continuous(breaks = seq(0, max(ylim), by = 1000),
                     minor_breaks = seq(0, max(ylim), by = 1000)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  scale_color_manual(values = cols) +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

Prediction intervals

# compute prediction using Moving Block Bootstrap (MBB) for nls
df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1))

pred1 = cbind(df, "fit" = predict(mod1, newdata = df))
pred1[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod1, df[df$x > max(data$x),])[,2:3]

pred2 = cbind(df, "fit" = predict(mod2, newdata = df))
pred2[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod2, df[df$x > max(data$x),])[,2:3]

pred3 = cbind(df, "fit" = predict(mod3, newdata = df))
pred3[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod3, df[df$x > max(data$x),])[,2:3]

pred4 = cbind(df, "fit" = predict(mod4, newdata = df))
pred4[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod4, df[df$x > max(data$x),])[,2:3]

# predictions for next day
pred = rbind(subset(pred1, x == max(data$x)+1, select = 2:5),
             subset(pred2, x == max(data$x)+1, select = 2:5),
             subset(pred3, x == max(data$x)+1, select = 2:5),
             subset(pred4, x == max(data$x)+1, select = 2:5))
print(pred, digits = 3)
##           date   fit   lwr   upr
## 54  2020-04-17 27642 22352 33029
## 541 2020-04-17 21461 20586 22041
## 542 2020-04-17 22319 22026 22633
## 543 2020-04-17 22406 22087 22808

ylim = c(0, max(pred2$upr, pred3$upr, na.rm=TRUE))
ggplot(data, aes(x = date, y = y)) + 
  geom_point() +
  geom_line(data = pred1, aes(x = date, y = fit, color = "Exponential")) +
  geom_line(data = pred2, aes(x = date, y = fit, color = "Logistic")) +
  geom_line(data = pred3, aes(x = date, y = fit, color = "Gompertz")) +
  geom_line(data = pred4, aes(x = date, y = fit, color = "Richards")) +
  geom_ribbon(data = pred1, aes(x = date, ymin = lwr, ymax = upr), 
              inherit.aes = FALSE, fill = cols[1], alpha=0.3) +
  geom_ribbon(data = pred2, aes(x = date, ymin = lwr, ymax = upr), 
              inherit.aes = FALSE, fill = cols[2], alpha=0.3) +
  geom_ribbon(data = pred3, aes(x = date, ymin = lwr, ymax = upr),
              inherit.aes = FALSE, fill = cols[3], alpha=0.3) +
  geom_ribbon(data = pred4, aes(x = date, ymin = lwr, ymax = upr),
              inherit.aes = FALSE, fill = cols[4], alpha=0.3) +
  coord_cartesian(ylim = c(0, max(ylim))) +
  labs(x = "", y = "Deceased", color = "Model") +
  scale_y_continuous(minor_breaks = seq(0, max(ylim), by = 1000)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  scale_color_manual(values = cols) +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

Modelling recovered

# create data for analysis
data = data.frame(date = COVID19$data,
                  y = COVID19$dimessi_guariti,
                                    dy = reldiff(COVID19$dimessi_guariti))
data$x = as.numeric(data$date) - min(as.numeric(data$date)) + 1
DT::datatable(data, options = list("pageLength" = 5))

Estimation

Exponential

mod1_start = lm(log(y) ~ x, data = data)
b = unname(coef(mod1_start))
start = list(th1 = exp(b[1]), th2 = b[2])
exponential <- function(x, th1, th2) th1 * exp(th2 * x)
mod1 = nls(y ~ exponential(x, th1, th2), data = data, start = start)
summary(mod1)
## 
## Formula: y ~ exponential(x, th1, th2)
## 
## Parameters:
##      Estimate Std. Error t value          Pr(>|t|)    
## th1 976.94410   97.47035   10.02 0.000000000000119 ***
## th2   0.07199    0.00212   33.96           < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1756 on 51 degrees of freedom
## 
## Number of iterations to convergence: 11 
## Achieved convergence tolerance: 0.000009196

Logistic

mod2 = nls(y ~ SSlogis(x, Asym, xmid, scal), data = data)
summary(mod2)
## 
## Formula: y ~ SSlogis(x, Asym, xmid, scal)
## 
## Parameters:
##        Estimate Std. Error t value Pr(>|t|)    
## Asym 54466.8284  1853.3129   29.39   <2e-16 ***
## xmid    44.9109     0.6316   71.11   <2e-16 ***
## scal     8.3180     0.2197   37.86   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 601.1 on 50 degrees of freedom
## 
## Number of iterations to convergence: 0 
## Achieved convergence tolerance: 0.000002559

Gompertz

mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data)
summary(mod3)
## 
## Formula: y ~ SSgompertz(x, Asym, b2, b3)
## 
## Parameters:
##           Estimate    Std. Error t value Pr(>|t|)    
## Asym 116190.906902   6882.377734   16.88   <2e-16 ***
## b2        8.845674      0.193063   45.82   <2e-16 ***
## b3        0.960758      0.001321  727.12   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 349.2 on 50 degrees of freedom
## 
## Number of iterations to convergence: 0 
## Achieved convergence tolerance: 0.000004237

Richards

richards <- function(x, th1, th2, th3) th1*(1 - exp(-th2*x))^th3
Loss  <- function(th, y, x) sum((y - richards(x, th[1], th[2], th[3]))^2) 
start <- optim(par = c(coef(mod2)[1], 0.001, 1), fn = Loss, 
               y = data$y, x = data$x)$par
names(start) <- c("th1", "th2", "th3")
mod4 = nls(y ~ richards(x, th1, th2, th3), data = data, start = start,
           # trace = TRUE, # algorithm = "port", 
           control = nls.control(maxiter = 1000))
summary(mod4)
## 
## Formula: y ~ richards(x, th1, th2, th3)
## 
## Parameters:
##          Estimate    Std. Error t value          Pr(>|t|)    
## th1 219800.978994  32546.315009   6.753 0.000000014587917 ***
## th2      0.021818      0.002069  10.547 0.000000000000026 ***
## th3      4.478330      0.215971  20.736           < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 327.4 on 50 degrees of freedom
## 
## Number of iterations to convergence: 16 
## Achieved convergence tolerance: 0.0000004053

Models comparison

models = list("Exponential model" = mod1, 
              "Logistic model" = mod2, 
              "Gompertz model" = mod3,
              "Richards model" = mod4)
tab = data.frame(loglik = sapply(models, logLik),
                 df = sapply(models, function(m) attr(logLik(m), "df")),
                 Rsquare = sapply(models, function(m) 
                                  cor(data$y, fitted(m))^2),
                 AIC = sapply(models, AIC),
                 AICc = sapply(models, AICc),
                 BIC = sapply(models, BIC))
sel <- apply(tab[,4:6], 2, which.min)
tab$"" <- sapply(tabulate(sel, nbins = length(models))+1, symnum,
                 cutpoints = 0:4, symbols = c("", "*", "**", "***"))
knitr::kable(tab)
loglik df Rsquare AIC AICc BIC
Exponential model -470.1214 3 0.9846237 946.2428 946.7326 952.1537
Logistic model -412.7955 4 0.9980828 833.5910 834.4244 841.4722
Gompertz model -384.0157 4 0.9992583 776.0313 776.8646 783.9125
Richards model -380.5883 4 0.9993418 769.1766 770.0099 777.0577 *** 
ggplot(data, aes(x = date, y = y)) + 
  geom_point() +
  geom_line(aes(y = fitted(mod1), color = "Exponential")) +
  geom_line(aes(y = fitted(mod2), color = "Logistic")) +
  geom_line(aes(y = fitted(mod3), color = "Gompertz")) +
  geom_line(aes(y = fitted(mod4), color = "Richards")) +
  labs(x = "", y = "Recovered", color = "Model") +
  scale_color_manual(values = cols) +
  scale_y_continuous(breaks = seq(0, coef(mod2)[1], by = 1000),
                     minor_breaks = seq(0, coef(mod2)[1], by = 500)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

last_plot() +
  scale_y_continuous(trans = "log10", limits = c(10,NA)) +
  labs(y = "Recovered (log10 scale)")

Predictions

Point estimates

df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1),
               fit1 = predict(mod1, newdata = df),
               fit2 = predict(mod2, newdata = df),
               fit3 = predict(mod3, newdata = df),
               fit4 = predict(mod4, newdata = df))
ylim = c(0, max(df[,-(1:3)]))
ggplot(data, aes(x = date, y = y)) + 
  geom_point() + 
  geom_line(data = df, aes(x = date, y = fit1, color = "Exponential")) +
  geom_line(data = df, aes(x = date, y = fit2, color = "Logistic")) +
  geom_line(data = df, aes(x = date, y = fit3, color = "Gompertz")) +
  geom_line(data = df, aes(x = date, y = fit4, color = "Richards")) +
  coord_cartesian(ylim = ylim) +
  labs(x = "", y = "Recovered", color = "Model") +
  scale_y_continuous(breaks = seq(0, max(ylim), by = 1000),
                     minor_breaks = seq(0, max(ylim), by = 1000)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  scale_color_manual(values = cols) +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

Prediction intervals

# compute prediction using Moving Block Bootstrap (MBB) for nls
df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1))

pred1 = cbind(df, "fit" = predict(mod1, newdata = df))
pred1[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod1, df[df$x > max(data$x),])[,2:3]

pred2 = cbind(df, "fit" = predict(mod2, newdata = df))
pred2[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod2, df[df$x > max(data$x),])[,2:3]

pred3 = cbind(df, "fit" = predict(mod3, newdata = df))
pred3[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod3, df[df$x > max(data$x),])[,2:3]

pred4 = cbind(df, "fit" = predict(mod4, newdata = df))
pred4[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod4, df[df$x > max(data$x),])[,2:3]

# predictions for next day
pred = rbind(subset(pred1, x == max(data$x)+1, select = 2:5),
             subset(pred2, x == max(data$x)+1, select = 2:5),
             subset(pred3, x == max(data$x)+1, select = 2:5),
             subset(pred4, x == max(data$x)+1, select = 2:5))
print(pred, digits = 3)
##           date   fit   lwr   upr
## 54  2020-04-17 47652 42666 53026
## 541 2020-04-17 40790 38788 42438
## 542 2020-04-17 41967 40968 42745
## 543 2020-04-17 42308 41415 43115

ylim = c(0, max(pred2$upr, pred3$upr, na.rm=TRUE))
ggplot(data, aes(x = date, y = y)) + 
  geom_point() +
  geom_line(data = pred1, aes(x = date, y = fit, color = "Exponential")) +
  geom_line(data = pred2, aes(x = date, y = fit, color = "Logistic")) +
  geom_line(data = pred3, aes(x = date, y = fit, color = "Gompertz")) +
  geom_line(data = pred4, aes(x = date, y = fit, color = "Richards")) +
  geom_ribbon(data = pred1, aes(x = date, ymin = lwr, ymax = upr), 
              inherit.aes = FALSE, fill = cols[1], alpha=0.3) +
  geom_ribbon(data = pred2, aes(x = date, ymin = lwr, ymax = upr), 
              inherit.aes = FALSE, fill = cols[2], alpha=0.3) +
  geom_ribbon(data = pred3, aes(x = date, ymin = lwr, ymax = upr),
              inherit.aes = FALSE, fill = cols[3], alpha=0.3) +
  geom_ribbon(data = pred4, aes(x = date, ymin = lwr, ymax = upr),
              inherit.aes = FALSE, fill = cols[4], alpha=0.3) +
  coord_cartesian(ylim = c(0, max(ylim))) +
  labs(x = "", y = "Recovered", color = "Model") +
  scale_y_continuous(breaks = seq(0, max(ylim), by = 5000),
                     minor_breaks = seq(0, max(ylim), by = 1000)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  scale_color_manual(values = cols) +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

Description of evolution

Positive cases and administered swabs

df = data.frame(date = COVID19$data,
                positives = c(NA, diff(COVID19$totale_casi)),
                swabs = c(NA, diff(COVID19$tamponi)))
df$x = as.numeric(df$date) - min(as.numeric(df$date)) + 1
# df$y = df$positives/df$swabs
df$y = df$positives/c(NA, zoo::rollmean(df$swabs, 2))
df = subset(df, swabs > 50)
# DT::datatable(df[,-4], )
ggplot(df, aes(x = date)) + 
  geom_point(aes(y = swabs, color = "swabs"), pch = 19) +
  geom_line(aes(y = swabs, color = "swabs")) +
  geom_point(aes(y = positives, color = "positives"), pch = 0) +
  geom_line(aes(y = positives, color = "positives")) +
  labs(x = "", y = "Number of cases", color = "") +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  scale_color_manual(values = palette()[c(2,1)]) +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

ggplot(df, aes(x = date, y = y)) + 
  geom_smooth(method = "loess", se = TRUE, col = "black") +
  geom_point(col=palette()[4]) + 
  geom_line(size = 0.5, col=palette()[4]) +
  labs(x = "", y = "% positives among admnistered swabs (two-day rolling mean)") +
  scale_y_continuous(labels = scales::percent_format(),
                     breaks = seq(0, 0.5, by = 0.05)) +
  coord_cartesian(ylim = c(0,max(df$y, na.rm = TRUE))) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

Hospitalized and ICU patients

df = data.frame(date = COVID19$data,
                hospital = c(NA, diff(COVID19$totale_ospedalizzati)),
                icu = c(NA, diff(COVID19$terapia_intensiva)))
df$x = as.numeric(df$date) - min(as.numeric(df$date)) + 1
ggplot(df, aes(x = date, y = hospital)) + 
  geom_smooth(method = "loess", se = TRUE, col = "black") +
  geom_point(col = "orange") + 
  geom_line(size = 0.5, col = "orange") +
  labs(x = "", y = "Change hospitalized patients") +
  coord_cartesian(ylim = range(df$hospital, na.rm = TRUE)) +
  scale_y_continuous(minor_breaks = seq(min(df$hospital, na.rm = TRUE),
                                        max(df$hospital, na.rm = TRUE), 
                                        by = 100)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

ggplot(df, aes(x = date, y = icu)) + 
  geom_smooth(method = "loess", se = TRUE, col = "black") +
  geom_point(col = "red2") + 
  geom_line(size = 0.5, col = "red2") +
  labs(x = "", y = "Change ICU patients") +
  coord_cartesian(ylim = range(df$icu, na.rm = TRUE)) +
  scale_y_continuous(minor_breaks = seq(min(df$icu, na.rm = TRUE), 
                                        max(df$icu, na.rm = TRUE), 
                                        by = 10)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))